Question

Building a Storage Container

You are being asked to design an enclosed water storage container. You are given a budget of
$100 to build the container. It has been determined that the best material to make the container
costs $10 per m². (In other words your surface can not cost more than $100.) What is the
largest amount of water your container can hold? You will need to explain to your boss the
reasoning because they have a business degree and do not really remember any of their
geometry work from school.

Answer 1

2973.5 liters

Explanation:

You are being asked to find the volume of the largest possible container that has a given surface area. The volume will depend on the shape. If shape is not specified, we know a sphere has the largest volume for a given area.

__

area

The available area is ...

($100) / ($10/m²) = 10 m²

diameter

The area of a sphere with diameter d is ...

A = πd²

Then the diameter for the given area is ...

10 m² = πd²

(10/π) m² = d² . . . . . divide by π

√(10 m²/π) = d ≈ 1.78412 m . . . . . . diameter of the largest sphere

volume

The volume of a sphere is given by ...

V = 4/3πr³ = (π/6)d³

For the diameter found above, the volume of the sphere will be ...

V = (π/6)(1.78412 m)³ ≈ 2.97354 m³

There are 1000 liters in a cubic meter, so the largest amount of water that can be contained is ...

2.97354 m³ × (1000 L/m³) = 2973.54 L

The largest amount of water the container can hold is about 2973.5 liters.